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08 Dec 2021, 04:45
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
75%
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Question Stats:
56% (01:57) correct
44%
(02:25)
wrong
based on 373
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What is the remainder when \(25 * (1331)^{2196}\) is divided by 13?
(A) 0
(B) 1
(C) 7
(D) 11
(E) 12
Are You Up For the Challenge: 700 Level Questions
(A) 0
(B) 1
(C) 7
(D) 11
(E) 12
Are You Up For the Challenge: 700 Level Questions
08 Dec 2021, 06:23
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Bunuel
\(25 * (1331)^{2196}\) is divided by 13?
25 will leave a remainder of (-1)
1331 will leave a remainder of 5
So equation will be
(-1)*(5)^2196
or
-1*(25)^1098 again 25 will leave a remainder of -1 but since power is even so remainder will be 1
this leaves us with remainder = -1*1 = -1
Remainder can never be -ve so -1+13 = 12. E
General Discussion
08 Dec 2021, 05:17
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We can use the multiplicative rule of remainders to solve this question -
Remainder of (\(\frac{25}{13}\)) = 12
Remainder of (\(\frac{(1331) ^ {2196} }{13}\)) = (\(\frac{((13 * 102) + 5) ^ {2196} }{13}\))
Lets sub-divide the problem -
Remainder of (\(\frac{((13 * 102) ^ {2196} }{13}\)) = 0
So essentially we have to find out the remainder of (\(\frac{(5) ^ {2196} }{13}\))
= (\(\frac{(5) ^ {4 * 549} }{13}\))
= (\(\frac{((5) ^ {4}) ^{549} }{13}\))
Remainder of \(\frac{(5) ^ {4} }{13}\) = 1
Therefore (\(\frac{(5) ^ {4 * 549} }{13}\)) = (\(\frac{(1) ^ {549} }{13}\)) = 1
Ans = 12 * 1 = 12
IMO - E
Remainder of (\(\frac{25}{13}\)) = 12
Remainder of (\(\frac{(1331) ^ {2196} }{13}\)) = (\(\frac{((13 * 102) + 5) ^ {2196} }{13}\))
Lets sub-divide the problem -
Remainder of (\(\frac{((13 * 102) ^ {2196} }{13}\)) = 0
So essentially we have to find out the remainder of (\(\frac{(5) ^ {2196} }{13}\))
= (\(\frac{(5) ^ {4 * 549} }{13}\))
= (\(\frac{((5) ^ {4}) ^{549} }{13}\))
Remainder of \(\frac{(5) ^ {4} }{13}\) = 1
Therefore (\(\frac{(5) ^ {4 * 549} }{13}\)) = (\(\frac{(1) ^ {549} }{13}\)) = 1
Ans = 12 * 1 = 12
IMO - E
